MATH235 MATH235 Week 5 - Assessed problems (coursework) MATH235 Week 6 - Workshop problems

MATH235 Week 5 - Moodle Quiz-assessed problems

The data set on the CPI (consumer price index) in four western economies was introduced in Question Sheet 2. We will fit a regression model to these data, to see whether the CPI’s of France ( $x_{i,1}$ ) and Germany ( $x_{i,2}$ ) can be used to explain variability in the CPI of the United States ( $Y_{i}$ ),

{}\mathbb{E}[Y_{i}]=\beta_{1}+\beta_{2}x_{i,1}+\beta_{3}x_{i,2}

Although Table 3 shows only 10 years of data, in fact 20 years of data (1991–2010) are available and can be loaded into R from the file cpi. You should use this larger data set to answer the following questions.

Q5.1

Least squares estimates
Fit model CP1 using the function lm. The least squares estimates are

(a)

$\hat{\beta}=(-51.6,0.00401,1.52)$
(b)

$\hat{\beta}=(-40.5,0.610,0.778)$
(c)

$\hat{\beta}=(-50.8,0.0595,1.45)$
(d)

$\hat{\beta}=(-51.6,1.52,0.00401)$
(e)

$\hat{\beta}=(-50.8,1.45,0.0595)$

[marks: 2]

Q5.2

Evidence for relationships
95% confidence intervals for $\beta_{2}$ and $\beta_{3}$ are $(1.17,1.86)$ and $(-0.329,0.338)$ . From these we can conclude that,

(a)

At the 5% level, there is evidence of a relationship between the French and US CPI’s. There is also evidence of a relationship between the German and US CPI’s.
(b)

At the 5% level, there is no evidence of a relationship between the French and US CPI’s; neither is there evidence of a relationship between the German and US CPI’s.
(c)

At the 5% level, there is evidence of a relationship between the French and US CPI’s. There is no evidence of a relationship between the German and US CPI’s.
(d)

At the 5% level, there is no evidence of a relationship between the French and US CPI’s. However, there is evidence of a relationship between the German and US CPI’s.
(e)

At the 10% level, there is evidence of a relationship between the French and US CPI’s. There is no evidence of a relationship between the German and US CPI’s.

[marks: 2]

Q5.3

Confidence interval
Fitting the linear model for the US CPI with an intercept term and the German CPI as the only covariate gives

{}\mathbb{E}[Y_{i}]=-46.8+1.46x_{i,2}.

The standard error for the regression coefficient for $x_{i,2}$ is 0.505. What is the 95% confidence interval for this coefficient?

(a)

$(0.399,2.52)$
(b)

$(-0.0330,2.95)$
(c)

$(-0.329,0.338)$
(d)

$(0.924,2.00)$
(e)

$(0.395,2.47)$

[marks: 2]

Q5.4

Correlation
What is the correlation between the French and German CPI’s?

(a)

0.901
(b)

0.981
(c)

0.991
(d)

79.9
(e)

188.9

[marks: 2]

Q5.5

Contradiction
Is there a contradiction between the inferences to be drawn for the relationship between the German and US CPI’s from models CP1 and CP2. If so, use your answer to Q5.4 to explain why this contradiction comes about.

(a)

There is no apparent contradiction.
(b)

Model CP1 suggests no significant relationship between the German and US CPI’s, whereas model CP2 does. This is because the French and German CPI’s are collinear.
(c)

Model CP1 suggests no significant relationship between the German and US CPI’s, whereas model CP2 does. This is because the French and German CPI’s are orthogonal.
(d)

Model CP1 suggests a significant relationship between the German and US CPI’s, whereas model CP2 does not. This is because the French and German CPI’s are orthogonal.
(e)

Model CP1 suggests a significant relationship between the German and US CPI’s, whereas model CP2 does not. This is because the French and German CPI’s are collinear.

[marks: 2]